U.S. patent application number 12/100211 was filed with the patent office on 2009-10-15 for automated mud slowness estimation.
This patent application is currently assigned to SCHLUMBERGER TECHNOLOGY CORPORATION. Invention is credited to Hugues A. Djikpesse, Bikash K. Sinha, Henri-Pierre Valero.
Application Number | 20090257307 12/100211 |
Document ID | / |
Family ID | 41162593 |
Filed Date | 2009-10-15 |
United States Patent
Application |
20090257307 |
Kind Code |
A1 |
Valero; Henri-Pierre ; et
al. |
October 15, 2009 |
AUTOMATED MUD SLOWNESS ESTIMATION
Abstract
An integrated framework is described for automating some or all
of mud slowness estimation for both fast and slow formations. An
estimation of fluid slowness based on monopole radial profiling is
calculated if conditions permit. Alternatively, an estimation of
fluid slowness based on Scholte wave slowness is estimated if
conditions do not permit calculation based on monopole radial
profiling. Tool standoff may also be estimated based on monopole
radial profiling.
Inventors: |
Valero; Henri-Pierre;
(Belmont, MA) ; Djikpesse; Hugues A.; (Cambridge,
MA) ; Sinha; Bikash K.; (Cambridge, MA) |
Correspondence
Address: |
SCHLUMBERGER-DOLL RESEARCH;ATTN: INTELLECTUAL PROPERTY LAW DEPARTMENT
P.O. BOX 425045
CAMBRIDGE
MA
02142
US
|
Assignee: |
SCHLUMBERGER TECHNOLOGY
CORPORATION
Cambridge
MA
|
Family ID: |
41162593 |
Appl. No.: |
12/100211 |
Filed: |
April 9, 2008 |
Current U.S.
Class: |
367/31 ; 181/102;
324/324; 340/856.4; 367/25 |
Current CPC
Class: |
G01V 1/50 20130101 |
Class at
Publication: |
367/31 ; 367/25;
181/102; 324/324; 340/856.4 |
International
Class: |
G01V 1/50 20060101
G01V001/50; E21B 47/14 20060101 E21B047/14 |
Claims
1. A method of estimating an indication of fluid slowness
comprising: calculating an indication of fluid slowness based on
monopole radial profiling if conditions permit; and calculating an
indication of fluid slowness based on Scholte wave slowness if
conditions do not permit calculation based on monopole radial
profiling.
2. The method of claim 1 including the further step of determining
whether conditions permit calculation based on monopole radial
profiling output.
3. The method of claim 2 including the further step of determining
that conditions permit calculation based on monopole radial
profiling if formation slowness is fast or intermediate.
4. The method of claim 1 including the further step of calculating
an indication of tool standoff relative to borehole wall based on
monopole radial profiling output.
5. The method of claim 1 including the further step of utilizing
the Scholte slowness to determine an a priori distribution function
to be used to estimate mud slowness.
6. The method of claim 1 including the further step of calculating
a value corresponding to fluid slowness based on Scholte wave
slowness from Stoneley and dipole modes.
7. The method of claim 1 including the further step of preparing
for the calculating steps by estimating compressional slowness,
shear slowness, borehole fluid mass density and formation bulk
density from borehole data.
8. The method of claim 1 including the further step of calculating
a value corresponding to fluid slowness based on fastest Scholte
wave slowness if the formation is anisotropic.
9. The method of claim 1 including the further step of outputting
fluid slowness and relative uncertainty if calculating the
indication of fluid slowness based on monopole radial
profiling.
10. The method of claim 1 including the further step of inverting
travel time if calculating the indication of fluid slowness based
on monopole radial profiling.
11. Apparatus for estimating an indication of fluid slowness
comprising: at least one acoustic sensor for obtaining monopole
radial profiling data and Scholte wave slowness data; processing
circuitry for calculating an indication of fluid slowness based on
monopole radial profiling if conditions permit; and processing
circuitry for calculating an indication of fluid slowness based on
Scholte wave slowness if conditions do not permit calculation based
on monopole radial profiling.
12. The apparatus of claim 11 including processing circuitry that
determines whether conditions permit calculation based on monopole
radial profiling.
13. The apparatus of claim 12 including processing circuitry that
determines that conditions permit calculation based on monopole
radial profiling if formation slowness is fast or intermediate.
14. The apparatus of claim 11 including processing circuitry that
calculates an indication of tool standoff on monopole radial
profiling output.
15. The apparatus of claim 11 including processing circuitry that
utilizes the Scholte slowness to determine an a priori probability
distribution function.
16. The apparatus of claim 11 including processing circuitry that
calculates an indication of fluid slowness based on Scholte wave
slowness from Stoneley and dipole modes.
17. The apparatus of claim 11 including processing circuitry that
prepares for the calculations by estimating compressional slowness,
shear slowness, borehole fluid mass density and formation bulk
density from borehole data.
18. The apparatus of claim 11 including processing circuitry that
calculates an indication of fluid slowness based on fastest Scholte
wave slowness if the formation is anisotropic.
19. The apparatus of claim 11 including processing circuitry that
provides fluid slowness uncertainty if calculating the indication
of fluid slowness based on monopole radial profiling.
20. The apparatus of claim 11 including processing circuitry that
inverts travel time if calculating the indication of fluid slowness
based on monopole radial profiling
21. A computer readable medium including a computer program for
estimating an indication of fluid slowness comprising: logic for
calculating an indication of fluid slowness based on monopole
radial profiling if conditions permit; and logic for calculating an
indication of fluid slowness based on Scholte wave slowness if
conditions do not permit calculation based on monopole radial
profiling.
22. The computer readable medium of claim 21 including logic for
determining whether conditions permit calculation based on monopole
radial profiling as a function of formation slowness.
23. The computer readable medium of claim 22 including logic for
determining that conditions permit calculation based on monopole
radial profiling if formation slowness is fast or intermediate.
24. The computer readable medium of claim 21 including logic for
calculating an indication of tool standoff relative to borehole
wall on monopole radial profiling.
25. The computer readable medium of claim 21 including logic for
utilizing the Scholte slowness to determine an a priori
distribution function to be used to estimate the mud slowness
value.
26. The computer readable medium of claim 21 including logic for
calculating an indication of fluid slowness based on Scholte wave
slowness from Stoneley and dipole modes.
27. The computer readable medium of claim 21 including logic for
preparing for the calculations by estimating compressional
slowness, shear slowness, borehole fluid mass density and formation
bulk density from borehole data.
28. The computer readable medium of claim 21 including logic for
calculating an indication of fluid slowness based on fastest
Scholte wave slowness if the formation is anisotropic.
29. The computer readable medium of claim 21 including logic for
outputting fluid slowness uncertainty if calculating the indication
of fluid slowness based on monopole radial profiling.
30. The computer readable medium of claim 21 including logic for
inverting travel time if calculating the indication of fluid
slowness based on monopole radial profiling.
Description
BACKGROUND OF THE INVENTION
[0001] 1. Field of the Invention
[0002] This invention is generally related to acoustic data
analysis, and more particularly to automated estimation of fluid
slowness to facilitate acoustic logging and analysis
[0003] 2. Background of the Invention
[0004] Formations are characterized in terms of slowness values.
For example, a formation may be characterized as being slow if the
shear slowness, i.e., inverse of velocity, of the formation is
greater than the mud slowness. If the shear slowness of the
formation is less than the mud slowness then the formation may be
characterized as being fast. As described in Cheng, C. H., and
Toksoz, M. N., 1981, Elastic wave propagation in a fluid filled
borehole and synthetic acoustic logs, Geophysics, 46, p. 1042, in
fast formations it is known to utilize a monopole source, where
refracted compressional arrival time, refracted shear arrival time,
and a Stoneley wave which is guided by the fluid-rock interface are
excited by the monopole source. These various arrivals are usually
used to estimate, respectively, compressional, shear and Stoneley
slowness of the formation. As described in Paillet, F. L. and
Chang, C. H., 1991, Acoustic waves in borehole: CRC Press Inc, ISBN
0-8493-8890-2, Boca Raton, Ann Arbor, Boston, London, it is also
possible to use other modes such as leaky modes to get an estimate
of compressional slowness in a slow formation. However, in slow
formations it is not possible to measure formation shear slowness
from headwaves because shear waves do not exist in slow formations.
It is known to use a dipole transmitter that excites dipole
flexural waves in the borehole in order to overcome this
limitation. Like other borehole modes, the dipole mode is
dispersive (See Sinha, B. K. and Zeroug, S., 1997, Geophysical
prospecting using sonics and ultrasonics: Wiley Encyclopedia of
Electrical and Electronic Engineers, John G. Webster, Editor, John
Wiley and Sons, Inc.). However, it is possible to estimate
formation shear slowness by extracting the dipole slowness at low
frequencies as described by Kimball, C. V, and Marzetta, T. L.,
1987, Semblance processing of borehole acousticg data, Geophysics,
49, 530-544.
[0005] One factor that affects acoustic wave propagation
measurements in a fluid filled borehole is the fluid slowness,
e.g., mud slowness, where mud is disposed between the tool and the
borehole wall. There is no practical technique for measurement of
the mud slowness in a well at sonic frequencies. Various indirect
and direct evaluation techniques are known. However, these
techniques have some drawbacks.
[0006] Indirect evaluation of mud slowness can be based on
examination of mud samples at the surface or data from the
manufacturer of the mud components. However, these techniques tend
to be inaccurate because mud slowness is a function of conditions
which can differ significantly between the surface and locations of
interest within the well, e.g., pressure, temperature,
presence/absence of gas, etc. Empirical equations have been
developed that describe some common mud types, but errors can still
occur if incorrect assumptions about conditions are used, or if the
uncertainties of some parameters are too large.
[0007] Direct evaluation of mud slowness can be based on the
dispersive characteristics of some modes using a Prony-based method
as described by Lang, S. W., Kurkjian, A. L., McClellan, J. H.,
Morris, C. F., and Parks, T. W., 1987, Estimating slowness
dispersion array from arrays of sonic waveforms: Geophysics, 52
(4), 530-544. The technique involves transforming an array of time
waveforms into a frequency slowness domain to enable evaluation of
the characteristics of the various dispersive and non-dispersive
modes present in the recorded data, as described by Plona, T.,
Sinha, S., Kane, M., Bose, S., Wang, C., Pabon, J., Zeroug, S.,
2004, Identifying formation response using sonic dispersion curves,
74th Annual International Meeting of the Society of Exploration
Geophysicists (SEG), Denver, Expanded Abstracts. Various options
are available for performing this analysis, depending on the
formation type and modes considered. One option includes adjusting
the mud slowness in the modeling parameters to match the Stoneley
dispersion curve model to the dispersion curve computed from the
data. Another option is based on the fact that mud slowness is
asymptotically approached by both the Stoneley and flexural data.
The asymptote of the Stoneley dispersion curve at high frequency
must be slower or equal to the mud value, while the value of the
shear asymptotes must be faster than the mud (unless the formation
is damaged). The dipole-flexural curve converges to the Scholte
slowness, which is dependent on both mud slowness and the formation
properties close to the borehole wall. Another option is based on
the Leaky P mode. However, this option is only valid when a leaky
compressional is present in the data, i.e., in a slow formation.
The leaky modes can be considered as multiple reflected and
constructively interfering waves propagating in the borehole, as
described by Tichelaar, B. W. and Luik K. W., 1995, Sonic logging
of compressional-wave velocities in a very slow formation,
Geophysics, 60, 1627-1633; and Valero, H. P., Peng, L., Yamamoto,
M., Plona, T., Murray, D., Yamamoto, H., 2004, Processing of
monopole compressional in slow formation, 74th Annual International
Meeting of the Society of Exploration Geophysicists (SEG), Denver,
Expanded Abstracts. Unlike the refracted P head wave, leaky modes
are dispersive, i.e., starting at the compressional velocity at low
frequency and tending to the mud velocity as frequency increases.
Further, there exists a cutoff frequency below which they are not
excited. Although such dispersion analysis may be used to estimate
mud slowness, the technique requires time-consuming analysis of
various frames by skilled personnel. Further, none of the
techniques is suitable for all formations.
SUMMARY OF THE INVENTION
[0008] In accordance with an embodiment of the invention, a method
of estimating an indication of fluid slowness comprises calculating
an indication of fluid slowness based on monopole radial profiling
if conditions permit; and calculating an indication of fluid
slowness based on Scholte wave slowness if conditions do not permit
calculation based on monopole radial profiling.
[0009] In accordance with another embodiment of the invention,
apparatus for estimating an indication of fluid slowness comprises
at least one acoustic sensor for obtaining monopole radial
profiling data and Scholte wave slowness data; processing circuitry
for calculating an indication of fluid slowness based on monopole
radial profiling if conditions permit; and processing circuitry for
calculating an indication of fluid slowness based on Scholte wave
slowness if conditions do not permit calculation based on monopole
radial profiling. It should also be noted that the mud slowness
estimated from the Scholte wave slowness can be used as a priori
information for a probability distribution function (PDF) of the
fluid while calculating based on the monopole radial profiling
technique.
[0010] An advantage of the invention is that it helps provide an
integrated framework capable of automating some or all of mud
slowness estimation for both fast and slow formations. A first
analysis, suitable for fast formations, is based on a probabilistic
approach using high frequency monopole data. In particular, the
results of the monopole radial image are used to obtain an
estimation of the mud slowness. The second analysis, suitable for
slow formations, is based on the use of the Scholte wave slowness.
The second analysis depends on mud and formation material
properties, but is independent of borehole radius. Both techniques
are combined in one unified and automated framework to facilitate
automated operation in both fast and slow formations.
[0011] Further features and advantages of the invention will become
more readily apparent from the following detailed description when
taken in conjunction with the accompanying drawings.
BRIEF DESCRIPTION OF THE DRAWINGS
[0012] The present invention is further described in the detailed
description which follows, in reference to the noted plurality of
drawings by way of non-limiting examples of exemplary embodiments
of the present invention, in which like reference numerals
represent similar parts throughout the several views of the
drawings, and wherein:
[0013] FIG. 1 illustrates a wellsite system in which the present
invention can be employed, and a LWD embodiment, according to
embodiments of the invention;
[0014] FIG. 2 illustrates a wireline logging embodiment, according
to embodiments of the invention;
[0015] FIG. 3 illustrates automated estimation of fluid slowness
and tool standoff.
[0016] FIG. 4 illustrates the MRP technique in greater detail,
according to embodiments of the invention;
[0017] FIG. 5 illustrates an example of lognormal distribution for
water based mud, according to embodiments of the invention;
[0018] FIG. 6 illustrates mud compressional slowness estimation
using Stoneley dispersion at high frequencies or STC processing of
the high-frequency Stoneley data where it is nearly non-dispersive,
according to embodiments of the invention;
[0019] FIG. 7 illustrates an embodiment of the overall integrated
technique in greater detail, according to embodiments of the
invention;
DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS
[0020] The particulars shown herein are by way of example and for
purposes of illustrative discussion of the embodiments of the
present invention only and are presented in the cause of providing
what is believed to be the most useful and readily understood
description of the principles and conceptual aspects of the present
invention. In this regard, no attempt is made to show structural
details of the present invention in more detail than is necessary
for the fundamental understanding of the present invention, the
description taken with the drawings making apparent to those
skilled in the art how the several forms of the present invention
may be embodied in practice. Further, like reference numbers and
designations in the various drawings indicated like elements.
[0021] FIG. 1 illustrates a wellsite system in which the present
invention can be employed. The wellsite can be onshore or offshore.
In this exemplary system, a borehole (11) is formed in subsurface
formations by rotary drilling in a manner that is well known. A
drill string (12) is suspended within the borehole (11) and has a
bottom hole assembly (100) which includes a drill bit (105) at its
lower end. The surface system includes platform and derrick
assembly (10) positioned over the borehole (11), the assembly (10)
including a rotary table (16), kelly (17), hook (18) and rotary
swivel (19). The drill string (12) is rotated by the rotary table
(16), energized by means not shown, which engages the kelly (17) at
the upper end of the drill string. The drill string (12) is
suspended from a hook (18), attached to a traveling block (also not
shown), through the kelly (17) and a rotary swivel (19) which
permits rotation of the drill string relative to the hook. As is
well known, a top drive system could alternatively be used.
[0022] The surface system may include drilling fluid or mud (26)
stored in a pit (27) formed at the well site. A pump (29) delivers
the drilling fluid (26) to the interior of the drill string (12)
via a port in the swivel (19), causing the drilling fluid to flow
downwardly through the drill string (12) as indicated by the
directional arrow (8). The drilling fluid exits the drill string
(12) via ports in the drill bit (105), and then circulates upwardly
through the annulus region between the outside of the drill string
and the wall of the borehole, as indicated by the directional
arrows (9). In this well known manner, the drilling fluid
lubricates the drill bit (105) and carries formation cuttings up to
the surface as it is returned to the pit (27) for
recirculation.
[0023] Acoustic data is gathered with a logging tool which may be
of any type, including but not limited to wireline type (See FIG.
2) and logging while drilling type (See FIG. 1). The bottom hole
assembly (100) of the embodiment illustrated in FIG. 1 includes a
logging-while-drilling (LWD) module (120), a
measuring-while-drilling (MWD) module (130), a roto-steerable
system and motor, and drill bit (105). The LWD module (120) is
housed in a special type of drill collar, as is known in the art,
and can contain one or a plurality of known types of logging tools.
It will also be understood that more than one LWD and MWD module
can be employed, e.g. as represented at (120A). (References,
throughout, to a module at the position of (120) can alternatively
mean a module at the position of (120A) as well.) The LWD module
includes capabilities for measuring, processing, and storing
information, as well as for communicating with the surface
equipment. In the present embodiment, the LWD module includes a
pressure measuring device. The MWD module (130) is also housed in a
special type of drill collar, as is known in the art, and can
contain one or more devices for measuring characteristics of the
drill string and drill bit. The MWD tool further includes an
apparatus (not shown) for generating electrical power to the
downhole system. This may typically include a mud turbine generator
powered by the flow of the drilling fluid, it being understood that
other power and battery systems may be employed. In the present
embodiment, the MWD module includes one or more of the following
types of measuring devices: a weight-on-bit measuring device, a
torque measuring device, a vibration measuring device, a shock
measuring device, a stick slip measuring device, a direction
measuring device, and an inclination measuring device.
[0024] Referring to FIG. 2, a wireline logging embodiment includes
a logging tool (106) suspended from an armored cable (108), and may
have optional centralizers (not shown). The cable (108) extends
from the borehole (104) over a sheave wheel (110) on a derrick
(112) to a winch forming part of surface equipment, which may
include an analyzer unit (114). Well known depth gauging equipment
(not shown) may be provided to measure cable displacement over the
sheave wheel (110). The tool (106) may include any of many well
known devices to produce a signal indicating tool orientation.
Processing and interface circuitry within the tool (106) amplifies,
samples and digitizes the tool's information signals for
transmission and communicates them to the analyzer unit (114) via
the cable (108). Electrical power and control signals for
coordinating operation of the tool (106) may be generated by the
analyzer unit (114) or some other device, and communicated via the
cable (108) to circuitry provided within the tool (106). The
surface equipment includes a processor subsystem (116) (which may
include a microprocessor, memory, clock and timing, and
input/output functions--not separately shown), standard peripheral
equipment (not separately shown), and a recorder (118). The logging
tool (106) is representative of any logging device that may be used
in accordance with principles described herein. It will be
understood by those of skill in the art having the benefit of this
disclosure that other suitable logging device, including LWD
devices, can also be utilized.
[0025] Referring now to both FIGS. 1 and 2, the logging tool,
regardless of type, includes at least one acoustic transmitter
(109) and at least one acoustic receiver (126). The transmitter is
able to excite monopole and dipole acoustic modes. The illustrated
logging tool may also include multi-pole transmitters such as
crossed dipole transmitters and monopole transmitters capable of
exciting compressional, shear, Stoneley, and flexural waves. In the
illustrated example a plurality of receivers are arranged on the
logging tool at different spacing from the transmitters. The use of
a plurality of receivers and transmitters results in improved
signal quality and adequate extraction of the various borehole
signals over a wide frequency band. The distances and numbers of
receivers and transmitters shown in this example are not intended
to be limiting.
[0026] FIG. 3 illustrates a method by which the logging tool may be
utilized to help automate estimation of fluid slowness and tool
standoff from the borehole wall. In an initial step (300), a
determination is made as to whether Monopole Radial Profiling (MRP)
is a practical technique to apply. The outcome of step (300) is a
radial velocity profile around the borehole, including primarily
formation slowness. If conditions are conducive to use of MRP,
e.g., in the case of fast and intermediate formations, then MRP is
utilized as shown in step (302) to estimate both tool standoff and
fluid velocity as shown in steps (308, 310)). If conditions are not
conducive to use of MRP, e.g., in the case of slow formations, then
dipole/Stoneley dispersion curves are used to estimate fluid
slowness as shown in step (304). In particular, the dipole/Stoneley
dispersion curves may be utilized to estimate fluid velocity based
on Scholte slowness as shown in step (306). However, even in the
case of a fast/intermediate formation, MRP is utilized to estimate
tool standoff as shown in step (308). Further, even in the case of
a fast/intermediate formation, Scholte slowness from the
dipole/Stoneley dispersion curves may be used to set an a priori
distribution function. Characteristics of slow, intermediate and
fast formations are provided in U.S. Pat. No. 6,654,688, entitled
PROCESSING SONIC WAVEFORM MEASUREMENTS, which is incorporated by
reference.
[0027] FIG. 4 illustrates the MRP technique in greater detail. The
first arrival times measured for the different transmitter-receiver
spacings as shown in step (400) are used to estimate the posterior
probability density function (PDF) (402) associated with the mud
velocity v.sub.f and standoff s under the assumption that the
borehole and tool related parameters are known. The posteriori PDF
is used to estimate fluid slowness with uncertainty and tool
standoff with uncertainty as shown in steps (410, 412). In addition
to the unknown mud velocity and standoff, the uncertain parameters
are the thickness H.sub.j and the velocity V.sub.j of the layers
surrounding the borehole. For a given layer indexed j, the layer
thicknesses and velocities (H.sub.j,V.sub.j) and their probability
distributions are estimated (404) from the differential transit
times as described by Valero, H. P., Zeroug, S., Bose, S., A
ray-based sonic DTC radial profiling algorithm, SDR Research Note,
OFSR/RN/2005/090/MM/C, 2005, and published patent application
US-2006-0233047-A1, both of which are incorporated by reference.
The uncertainty associated with H.sub.j and V.sub.j is assumed
described by a random variable that is Gaussianly distributed with
mean and standard deviation .sigma..sub.H.sub.j,.sigma..sub.V.sub.j
respectively. Letting m represent the unknowns of the problem,
i.e., m=(v.sub.f,s) where v.sub.f is the fluid velocity and s the
standoff, d.sub.obs represents the first break of the compressional
head wave measured for different TR spacings and the pair (H,V)
describing the known but uncertain layer thicknesses and
velocities. The joint a posteriori probability distribution of any
model m to fit the observed first arrival times d.sub.obs and a
state of prior information I can be expressed as:
P ( m | d , I ) = P ( m | I ) P ( d | m , I ) P ( d | I ) ( 1 )
##EQU00001##
or when neglecting P(d|I), the prior probability density function
of the data
P(m|d,I).varies.P(m|I)P(d|m,I). (2)
In equations 1 and 2, P(m|I) the a-priori information describing
the knowledge of the model irrespective of the data and P(d|m,I) is
a likelihood function (408). The data vector d represents the
various predicted transit time for the given model m , i.e.,
d={t.sub.1,t.sub.2, . . . ,t.sub.m} (3)
[0028] Note that the expression of the transit time given an
altered formation is given by
TOF i = 2 s v f [ 1 - v f 2 V i 2 ] 1 / 2 + 2 j = 1 i - 1 H j V j [
1 - V j 2 V i 2 ] 1 / 2 + X V i ; i = 1 , K ; K .ltoreq. K max ( 4
) ##EQU00002##
Among these time of flight the shortest is assigned to X.sub.i that
is TOF=min[TOF,i=1,K.sub.i].
[0029] The a-priori probability distribution function (406) is
indicative of knowledge related to mud slowness. The probability is
associated with a range, e.g., mud slowness within the range [170
.mu.s/ft 300 .mu.s/ft] in the case of water based mud, although a
different range could be utilized. It is known that the probability
of occurrence of particular values of mud velocity is not uniform
in this interval and that, for most of the cases, the probable mud
slowness for a water based mud is around 200 .mu.s/ft. Similar
distribution functions are applicable to oil based and brine mud
slowness. Therefore, the choice for the a priori probability
function for the mud slowness distribution (406) is the lognormal
distribution function, however other distributions could be applied
too without changing the computational workflow of the invention.
As described by Rade, L., and Westergre, B., 2004, Mathematics
handbook for science and engineering, 5th ed, Springer, 562 pp, the
lognormal distribution function can be defined as:
f ( x ) = 1 ( x - .theta. ) .sigma. 2 .pi. exp [ - ln [ x - .theta.
m ] 2 .sigma. ] with x .gtoreq. .theta. ; m .gtoreq. 0 ; .sigma.
> 0. ( 5 ) ##EQU00003##
where .sigma.,.theta., and m are respectively the shape, location
and scale parameters. When .theta.=0 and m=1 it corresponds to the
standard lognormal distribution while when .theta.=0 it is called
the 2-parameter lognormal distribution. The expression for the
standard lognormal distribution is therefore:
f ( x ) = 1 x .sigma. 2 .pi. exp [ - ln [ x ] 2 .sigma. ] 2 with x
.gtoreq. 0 ; .sigma. > 0. ( 6 ) ##EQU00004##
[0030] FIG. 5 illustrates a lognormal distribution for water based
mud. In practice, the "center" of the distribution is set based on
some empirical calculation or on some a-priori knowledge from other
wells or other indirect measurements. The center distribution can
also be set using Scholte slowness. The standard deviation controls
the uncertainty and therefore is defined depending on the trust in
this a-priori information. Table 1 lists parameters associated with
water, oil and brine mud. Note however these parameters are merely
presented as examples, and are not intended to be limiting in any
way.
TABLE-US-00001 Mud Type Range .mu.s/ft Bean .mu.s/ft Standard
Deviation .mu.s/ft Water 170-300 200 20 Oil 220-270 250 20 Brine
160-190 175 20
[0031] An a priori probability distribution function may also be
defined for standoff s as shown in step (414, FIG. 4). According to
one caliper-based technique for measuring standoff, acceptable
accuracy of the measurement can be on the order of .+-.0.5'' for a
borehole of diameter of less than 19''. Assuming a 90% chance that
the accuracy is inferior or equal to 0.5'', which is equivalent to
having 0.5''=1.64.sigma., the uncertainty of the standoff is
0.5 1.64 or 0.3049 '' . ##EQU00005##
[0032] The likelihood function (408) measures how well the data
predicted by a given model m fits the observed measurements.
Equation 4 relating the data and model parameters is a non linear
relation that can be written in a matrix form as:
d=G(m) (7)
[0033] Note that the relations are non linear but can be linearized
locally. Under the assumption of Gaussian data uncertainty
distribution, the likelihood function can be expressed as
P ( d | m , I ) = K exp [ - 1 2 ( d - d obs ) T C D - 1 ( d - d obs
) ] with ( 8 ) K = [ ( 2 .pi. ) L det { C D } ] - 1 / 2 ( 9 )
##EQU00006##
[0034] where C.sub.D is the covariance matrix describing the
uncertainties related to the data. More precisely, C.sub.D combines
the uncertainties associated with the observed measurements (here
represented by the covariance matrix C.sub.d) and the ones
(C.sub.T) describing the errors related to the theoretical model
(i.e., the forward modeling). Since the covariance matrices C.sub.d
and C.sub.T describe uncertainty associated with random Gaussianly
distributed variables, the total uncertainty covariance matrix
C.sub.D is the sum of the individual matrices:
C.sub.D=C.sub.d+C.sub.T. (10)
[0035] When the errors of the theoretical model are assumed small
as compared to the ones associated to the measurements, it follows
that:
C.sub.D.apprxeq.C.sub.d. (11)
[0036] Two exemplary cases will now be described. First, with
C.sub.D being proportional to the matrix identity leading to the
simplification of equation 8 and 9 as
P ( d | m , I ) = K exp [ - 1 2 i = 1 L t i - t obs 2 .sigma. i 2 ]
. ( 12 ) ##EQU00007##
[0037] This equation represents the likelihood function assuming a
least square l.sub.2 norm of fit. If a Laplacian distribution is
considered to describe the uncertainties associated with the data,
the likelihood function based on the l.sub.1 norm would be:
P ( d | m , I ) = H exp [ i = 1 L t i - t obs ] . ( 13 )
##EQU00008##
[0038] FIG. 6 illustrates mud slowness estimation using Scholte
slowness from the Stoneley and dipole dispersion curves in greater
detail. Borehole fluid (mud) compressional slowness is used to
estimate formation shear modulus C.sub.66 and fluid mobility using
the borehole Stoneley data. The zero frequency intercept of
Stoneley dispersion yields the tube wave velocity, whereas the high
frequency Stoneley dispersion asymptotes to the Scholte slowness.
The Scholte wave is an interfacial non-dispersive wave that decays
away from the borehole surface. The Scholte wave slowness depends
on mud and formation material properties, but is independent of
borehole radius. In contrast, the lowest-order flexural dispersion
asymptotes to the formation shear slowness at low frequencies and
to the same Scholte slowness at high frequencies, provided the
formation is isotropic. Consequently, when the Scholte slownesses
from the high frequency asymptotes of the Stoneley and dipole
flexural dispersions are different, it is an indicator of
structural or stress-induced anisotropy in the data. The tube wave
velocity (velocity is inverse of slowness) can be used to estimate
the mud compressional velocity. However, measurement of tube wave
velocity can be challenging because of a lack of Stoneley signal
energy at very low frequencies.
[0039] Both the lowest-order axi-symmetric Stoneley and flexural
dispersions asymptote to the Scholte velocity (or slowness) at high
frequencies in an effectively isotropic formation. When the Scholte
wave velocity is known either from the Stoneley or flexural wave
data, we can estimate the mud compressional velocity v.sub.f using
the following equation from Norris, A. N. and Sinha, B. K., 1995,
The speed of a wave along a fluid-solide interface in the presence
of anisotropy and prestress, 1995 J. Acoust. Soc. Am., 98(2), pp.
1147-1154:
1 v f 2 = 1 V Sch 2 - .rho. r 2 V s 8 [ 4 V Sch 2 ( 1 V Sch 2 - 1 V
s 2 ) 1 2 - ( 2 V Sch 2 - 1 V s 2 ) 2 ( 1 V Sch 2 - 1 V P 2 ) - 1 2
] 2 where .rho. r = .rho. f .rho. b and V s 2 = C 66 .rho. b . ( 14
) ##EQU00009##
.rho..sub.f and .rho..sub.b denote the borehole fluid mass density
and formation bulk density, respectively; V.sub.P and V.sub.S are
the formation compressional and shear velocities, respectively;
C.sub.66 represents the shear modulus in the borehole
cross-sectional plane.
[0040] The procedure for obtaining the mud compressional slowness
includes several steps. First, at a given depth, estimate DTc, DTs,
.rho..sub.f and .rho..sub.b from borehole data as shown in step
(600). Second, estimate the Scholte slowness using the Stoneley
dispersion at high frequencies or from the STC processing of the
high-frequency Stoneley data where it is nearly non-dispersive as
shown in step (602). Third, use a Scholte slowness transform to
obtain the mud compressional slowness as shown in step (604). For
example, for a given formation with [0041] DTc=120 .mu.s/ft;
DTs=233.99 .mu.s/ft; .rho..sub.b=2500 kg/m.sup.3; mud density
.rho..sub.f=1450 kg/m.sup.3; an estimated Scholte slowness of 300
.mu.s/ft, yields a mud slowness of 250 .mu.s/ft based on a curve
representing the relationship between mud compressional slowness
and formation shear slowness for a fixed value of formation
compressional and Scholte slowness. When both the mud compressional
slowness and the Scholte slowness are known, it is possible to
estimate the formation shear slowness.
[0042] In the case of an anisotropic formation, e.g., TIV, shear
moduli C66, C.sub.44 and C.sub.55 are used to estimate the mud
slowness from Scholte wave slowness. Mud slowness may be computed
using equation but inputting into this equation the shear velocity
value computed respectively from C.sub.66 and C.sub.44 as if for an
isotropic formation. The result is two values for the mud slowness
called respectively mud.sub.C.sub.66 and mud.sub.C.sub.44. These
computed values become the initial guess for estimating the mud
slowness in an anisotropic formation. The true mud, i.e.,
mud.sub.aniso, will then be defined in the interval bounded by the
two mud values computed previously and can be defined as, for
example, the middle of this interval. Note that one can also use a
linear combination of both computed mud, i.e. C.sub.66 and C.sub.44
to get the final estimate of the mud in anisotropic formation. The
a priori will therefore be set using mud.sub.aniso and the width of
the distribution can be defined as half of the interval defined by
mud.sub.C.sub.66 and mud.sub.C.sub.44. Note finally that the
calculation done with C.sub.44 could also be done with C.sub.55.
Alternatively, because the Scholte wave slownesses for shear moduli
C.sub.66, C.sub.44 and C.sub.55 do not converge for an anisotropic
formation, calculation may be based on fastest Scholte wave
slowness, which will be closest to DT.sub.mud.
[0043] FIG. 7 illustrates an embodiment of the overall integrated
technique in greater details. The following explanation is for
estimation at one depth, but can be utilized for an interval by
applying the procedure at different depths. The first step (700) is
to estimate DTc,DTs, .rho..sub.f and .rho..sub.b from sonic data
(702). The second step (704) is to estimate the Scholte slowness
using the Stoneley dispersion at high frequencies or from the
standard STC processing of the high-frequency Stoneley data where
it is nearly non-dispersive, as shown in (706). The computed
Scholte slowness is then used to get an estimate of the mud
slowness (708). At this stage it is possible to calculate (710) the
type of formation, i.e., fast/intermediate or slow. In the case of
a slow formation, i.e., where a leaky compressional can provide an
estimate of the compressional slowness when present in the data,
then the final mud slowness estimated (712) will be the one
computed from the Scholte slowness procedure described above. In
the case of a fast formation, the a priori probability function is
set using the mud slowness estimate computed from the Scholte wave
as shown in step (714). This has the advantage of reducing the need
for external input from expert personnel. After setting the
parameters of the a-priori probability distribution function, the
likelihood is computed (716) as explained previously. Profiling and
transit time are obtained from the monopole data (718). The
a-posteriori probability is then computed (720) from the likelihood
and the a-priori probability distribution function. The mud
slowness is then estimated (722) from the a-posteriori probability
distribution with related uncertainties. Note that the integrated
framework is capable of automating some or all of mud slowness
estimation for both fast and slow formations.
[0044] While the invention is described through the above exemplary
embodiments, it will be understood by those of ordinary skill in
the art that modification to and variation of the illustrated
embodiments may be made without departing from the inventive
concepts herein disclosed. Moreover, while the preferred
embodiments are described in connection with various illustrative
structures, one skilled in the art will recognize that the system
may be embodied using a variety of specific structures.
Accordingly, the invention should not be viewed as limited except
by the scope and spirit of the appended claims.
* * * * *